Reconstruction algorithm
In this section, we present a fast and simple oneshot numerical reconstruction algorithm. Our numerical procedure is based on the asymptotic formula presented in Theorem 3.1, which describes the variation of the Kohn–Vogeliustype functional \(\mathcal{K}\) with respect to the creation of a small hole \(\omega _{z,\varepsilon }\) inside the domain Ω. From Theorem 3.1 it follows that the function \(\mathcal{K}\) satisfies the following topological asymptotic expansion:
$$\begin{aligned} \mathcal{K}(\Omega \backslash \overline{\omega _{z,\varepsilon }})= \mathcal{K}( \Omega )+ \frac{2\pi }{\log (\varepsilon )}\delta \mathcal{K}(z) + o\biggl(\frac{1}{\log (\varepsilon )} \biggr), \end{aligned}$$
where \(\delta \mathcal{K}\) is the topological gradient defined as
$$ \delta \mathcal{K}(x)=\gamma (x) \bigl( \bigl\vert \psi ^{d}_{0}(x) \bigr\vert ^{2} \bigl\vert \psi ^{n}_{0}(x) \bigr\vert ^{2}\bigr),\quad x\in \Omega , $$
with solutions \(\psi _{0}^{n}\) and \(\psi _{0}^{d}\) to the Neumann and Dirichlet problems (10)–(11).
According to the main idea of the topological sensitivity analysis method, the unknown plasma domain \(\Omega _{p}\) is likely to be located at the zone where the topological gradient \(\delta \mathcal{K}\) is negative. To present our proposed procedure, we introduce some notations. Let \(\delta _{\min }\) be the most negative value of the topological gradient \(\delta \mathcal{K}\) in Ω (i.e \(\delta _{\min }=\min_{x\in \Omega }\delta \mathcal{K}(x)\)). For all \(\rho \in [0,1]\), we define the zone
$$ \omega _{\rho }= \bigl\{ x\in \Omega ; \delta \mathcal{K}(x)\leq (1 \rho )\delta _{\min } \bigr\} . $$
The aim is reconstructing the location and shape of the unknown plasma domain \(\Omega _{p}\subset \Omega \) from a given measurement of the potential field on the boundary \(\Gamma =\partial \Omega \). The main steps of our numerical reconstruction procedure are summarized in the following “oneshot” algorithm.
Plasma reconstruction algorithm:

Step 1: Solve the problems (\(\mathcal{P}^{n}_{0}\)) and (\(\mathcal{P}^{d}_{0}\)),

Step 2: Compute the topological gradient \(\delta \mathcal{K}(z)\), \(z\in \Omega \),

Step 3: Reconstruct the plasma domain
$$ \Omega _{p}= \bigl\{ x\in \Omega ; \delta \mathcal{K}(x)=\bigl(1\rho ^{\star }\bigr) \delta _{\min }< 0 \bigr\} ,$$
where \(\rho ^{\star }\in \mathopen{]}0,1[\) is chosen to ensure the maximal \(\mathcal{K}\) such that
$$ \mathcal{K}(\Omega \backslash \overline{\omega _{\rho ^{\star }}}) \leq \mathcal{K}(\Omega \backslash \overline{\omega _{\rho }}),\quad \forall \rho \in (0,1).$$
In this oneiteration algorithm:

The location of the unknown plasma is given by the point \(z^{\star }\in \Omega \) where the topological gradient \(\delta \mathcal{K}\) is most negative (i.e., \(z^{\star }= \operatorname*{arg min}_{x\in \Omega }\delta \mathcal{K}(x)\)). In other words, the topological gradient tell us that the plasma domain is located around the point \(z^{\star }\in \Omega \).

The size of the domain occupied by the plasma is adjusted via the choice of the parameter \(\rho \in (0, 1)\). Its boundary is approximated by a levelset curve of the topological gradient \(\delta \mathcal{K}\) (i.e., the plasma domain is delimited by a wellchosen isovalue of \(\delta \mathcal{K}\))
$$ \partial \Omega _{p}= \bigl\{ x\in \Omega ; \delta \mathcal{K}(x)= \bigl(1 \rho ^{\star }\bigr)\delta _{\min } \bigr\} .$$

The computation of the plasma domain size (in Step 3) can be considered as a descent method where the step length is given by the volume variation \(V_{k}=\omega _{\rho _{k}}\backslash \omega _{\rho _{k+1}} \). The determination of \(\rho ^{\star }\) can be viewed as a line search step. However, determining the optimal value of the parameter ρ and optimizing the size of the domain ω solution to
$$ \min_{\rho \in (0, 1)}\mathcal{K}(\Omega \backslash \overline{\omega _{\rho }})$$
is still an open question. To speed up the convergence of our optimization algorithm, we perform a binary search approach (dichotomy method) for approaching as most as possible the best value of ρ starting from \(\rho =\frac{1}{2}\).
Remark 4.1
In the particular case where the exact plasma domain \(\Omega _{p}^{\operatorname{ex}}\) is known, the best value \(\rho ^{*}\) of the parameter ρ can be determined as the minimum of the following error functional:
$$\begin{aligned} \operatorname{er}(\rho )= \bigl[\operatorname{meas}\bigl(\Omega _{p}^{\operatorname{ex}}\cup \omega _{\rho }\bigr) \operatorname{meas}\bigl(\Omega _{p}^{\operatorname{ex}} \cap \omega _{\rho }\bigr) \bigr]/\operatorname{meas}\bigl(\Omega _{p}^{\operatorname{ex}} \bigr) \quad \forall \rho \in (0,1), \end{aligned}$$
(12)
where \(\operatorname{meas}(B)\) is the Lebesgue measure of a set \(B\subset \mathbb{R}^{2}\).
Numerical implementation
In our numerical implementation the measurements data \(\psi _{m}\) are synthetic, that is, generated by numerical simulations. More precisely, let \(\Omega _{p}^{\operatorname{ex}}\) be the exact plasma domain to be identified. The Dirichlet measurement data \(\psi _{m}\) on the boundary Γ are retrieved as \(\psi _{m}={\psi _{\operatorname{ex}}^{n}}_{\Gamma }\), where \(\psi _{\operatorname{ex}}^{n}\) is the solution to the Neumann problem (see (6)) in the domain \(\Omega \backslash \overline{\Omega _{p}^{\operatorname{ex}}}\)
$$ \textstyle\begin{cases}  \operatorname{div} (\gamma (x) \nabla \psi _{\operatorname{ex}}^{n}) = 0& \text{in } \Omega \backslash \overline{\Omega _{p}^{\operatorname{ex}}}, \\ \gamma (x) \nabla \psi _{\operatorname{ex}}^{n}{\mathbf{n }} = \Phi& \text{on } \Gamma , \\ \psi _{\operatorname{ex}}^{n} = 0& \text{on } \partial \Omega _{p}^{\operatorname{ex}}. \end{cases} $$
(13)
Here the exact plasma domain is used only for generating the measured data \(\psi _{m}\).
In the numerical simulation the vacuum vessel region Ω is defined by the disc with center \(x_{0}=(2, 0)\) and radius \(r=1\), i.e., \(\Omega =B(x_{0}, 1)\). The resolution of the boundary value problems (\(\mathcal{P}^{n}_{0}\)) and (\(\mathcal{P}^{d}_{0}\)) is based on a classical \(\mathbb{P}_{1}\) finite element method. The approximated solutions are computed using a uniform mesh on the boundary ∂Ω (defined by the circle \(\mathcal{C}(x_{0}, 1)\)) with step size \(h=\pi /300\approx 0.01\). The numerical procedure is implemented using the free software FreeFem++ (see http://www.freefem.org/ff++/).
Next, we present some reconstruction results showing the efficiency of the proposed oneshot algorithm.
Numerical simulations
Here we apply our oneshot numerical algorithm for reconstructing the plasma domain in different situations. We start our numerical investigation by the identification of an unknown elliptical plasma domain from boundary measured data (see Sect. 4.3.1). After that, we will discuss the influence of some numerical parameters on the accuracy of the reconstructed results: the effect of the location in Sect. 4.3.2, the effect of the size in Sect. 4.3.3, and the effect of the shape in Sect. 4.3.4.
Reconstruction results
In this section, we apply our oneshot algorithm for reconstructing an elliptical plasma domain from boundary measurement. The measured data \(\psi _{m}\) is generated using the ellipse \(\Omega ^{\operatorname{ex}}_{p}=B(x_{0},r_{1},r_{2})\) with center \(x_{0}=(2, 0)\) and radii \(r_{1}=0.4\) and \(r_{2}=0.5\).
The obtained numerical results are described in Figs. 1 and 2:
− In Fig. 1, we plot the isovalues (levelset curves) of the topological gradient \(\delta \mathcal{K}\). We can note that the zone where the topological gradient g is negative (the red region) nearly coincides with the exact plasma domain \(\Omega ^{\operatorname{ex}}_{p}\) (see Fig. 2). From this figure we can observe that the most negative value of \(\delta \mathcal{K}\) is given by \(\delta _{\min }=0.0092\), which is reached at the point \(z^{\star }=(1.9, 0)\).
− To compute the optimal value \(\rho ^{\star }\) of the parameter ρ and estimate the size of the reconstructed plasma domain, we study the variation of the error function, which allows us to deduce that the optimal value of ρ is given by \(\rho ^{\star }=0.12\), i.e., \(1\rho ^{\star }=0.88\). According to our algorithm, the reconstructed plasma domain is defined as
$$ \Omega _{p}= \bigl\{ x\in \Omega ; \delta \mathcal{K}(x)\leq \bigl(1 \rho ^{\star }\bigr)\delta _{\min } \bigr\} ,\quad \text{with } \rho ^{\star }=0.12 \text{ and } \delta _{\min }=0.0092.$$
− In Fig. 2, we show the exact (delimited by the red line) and reconstructed (delimited by the black line) plasma domains. We conclude here that our numerical algorithm provides an efficient reconstruction result in only one iteration.
Effect of the location
In this section, we discuss the effect of the plasma location on the reconstruction result. The aim is reconstructing a circular plasma \(\Omega ^{\operatorname{ex}}_{p}=z+ 0.2 B(0,1)\) located at the point \(z\in \Omega \). We present in Fig. 3 the isovalues of the topological gradient δK for different plasma locations \(z_{i}\), \(1\leq i\leq 4\). As we can observe here, the topological gradient detects the exact plasma location, i.e., the most negative region coincides with the exact plasma domain.
Effect of the size
In this section, we discuss the effect of the plasma size on the reconstruction result. We illustrate in Fig. 4 the isovalues of the topological gradient δK for different plasma sizes \(r_{i}\), \(1\leq i\leq 3\).
Effect of the shape
In this section, we discuss the effect of the plasma shape on the reconstruction result. The first test is devoted to an ellipticalshaped plasma. The second test is concerned with a rectangularshaped plasma. As we can observe in Fig. 5, the topological gradient gives a good approximation of the unknown plasma in both cases.